Periodic solution of a turbidostat model with impulsive state feedback control

In this paper, a turbidostat model with impulsive state feedback control is considered. We obtain sufficient conditions of the global asymptotical stability of the system without impulsive state feedback control. We also obtain that the system with impulsive state feedback control may have order one periodic solution, and the sufficient condition for existence and stability of order one periodic solution is gotten as well. For some special cases, it is shown that in the system an order two periodic solution may exist. Our results show that the control measure is effective and reliable.     

Periodic solution of a pest management Gompertz model with impulsive state feedback control

In this paper, a new model with two state impulses is proposed for pest management. According to different thresholds, an integrated strategy of pest management is considered, that is to say if the density of the pest population reaches the lower threshold \(h_1\) at which pests cause slight damage to the forest, biological control (releasing natural enemy) will be taken to control pests; while if the density of the pest population reaches the higher threshold \(h_2\) at which pests cause serious damage to the forest, both chemical control (spraying pesticide) and biological control (releasing natural enemy) will be taken at the same time. For the model, firstly, we qualitatively analyse its singularity. Then, we investigate the existence of periodic solution by successor functions and Poincaré-Bendixson theorem and the stability of periodic solution by the stability theorem for periodic solutions of impulsive differential equations. Lastly, we use numerical simulations to illustrate our theoretical results.     

Nonlinear analysis of a microbial pesticide model with?impulsive?state feedback control

In this paper, a new mathematical model for the entomopathogenic nematode attacking pests with impulsive state feedback control is considered. By using the Poincaré map, we obtain that the system with impulsive state feedback control has a periodic solution of order one. Sufficient conditions for existence and stability of the order one periodic solution are given. Specifically, the system has a singular order one periodic solution. In some cases, it is possible that the system may also have an order two periodic solution. Our results show that the control measure is effective and reliable.     

Hopf bifurcation in a delayed food-limited model with feedback control

In this paper, we consider a delayed food-limited model with feedback control. By regarding the delay as the bifurcation parameter and analyzing the corresponding characteristic equations, the linear stability of the system is discussed, and Hopf bifurcations are demonstrated. By the normal form and the center manifold theory, the explicit formulae are derived to determine the stability, direction and other properties of bifurcating periodic solutions. Finally, some examples are presented to verify our main results.     

Elastic-plastic dynamic behavior of a beam-column model

Summary An elastoplastic beam-column model with one degree of freedom subjected statically to an on-center or off-center axial compressive force is considered. Its dynamic behavior in the elastoplastic range, resulting from a transverse periodic acting force in resonance with the structure in the elastic range, is examined. The principal features are highlighted by a phase plane study.

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Sommario Si considera il modello a un grado di libertà di un'asta elastoplastica sollecitata staticamente da una forza assiale di compressione centrata o eccentrica. Si esamina il suo comportamento dinamico in campo elastoplastico per effetto di una forza trasversale periodica e in risonanza con la struttura in campo elastico, evidenziandone le principali caratteristiche attraverso lo studio nel piano delle fasi.

     

Periodic solution of the system with impulsive state feedback control

The order-1 periodic solution of the system with impulsive state feedback control is investigated. We get the sufficient condition for the existence of the order-1 periodic solution by differential equation geometry theory and successor function. Further, we obtain a new judgement method for the stability of the order-1 periodic solution of the semi-continuous systems by referencing the stability analysis for limit cycles of continuous systems, which is different from the previous method of analog of Poincarè criterion. Finally, we analyze numerically the theoretical results obtained.     

Chaos in a pendulum with feedback control

We study chaotic dynamics of a pendulum subjected to linear feedback control with periodic desired motions. The pendulum is assumed to be driven by a servo-motor with small inductance, so that the feedback control system reduces to a periodic perturbation of a planar Hamiltonian system. This Hamiltonian system can possess multiple saddle points with non-transverse homoclinic and/or heteroclinic orbits. Using Melnikov's method, we obtain criteria for the existence of chaos in the pendulum motion. The computation of the Melnikov functions is performed by a numerical method. Several numerical examples are given and the theoretical predictions are compared with numerical simulation results for the behavior of invariant manifolds.     

A dynamic model for a Timoshenko beam in an elastic-plastic state

Summary In this paper, equations which describe the propagation of elastic-plastic waves of combined stress resultants in a Timoshenko beam under symmetrical bending and tension or compression are derived. It is assumed that the beam material exhibits a weak work-hardening and strain-rate independent behaviour. The normality rule is postulate for the plastic strain rates of an infinitesimal volume element. The wave propagation is described by a hyperbolic system of differential equations, the state variables of which are the stress resultants and the strain rates of an infinitesimal beam element. It is shown that the actual distribution of shear stress in the plastic range need not be taken into account by means of a plastic shear correction factor.

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Ein dynamisches Modell für einen Timoshenko-Balken im elastisch-plastischen Zustand

Übersicht Diese Arbeit stellt ein Modell vor, das die Ausbreitung von elastischen und plastischen Wellen kombinierter Schnittgrößen in dynamisch hochbeanspruchten Balken unter gerader Biegung und Normalkraft beschreibt. Das Werkstoff-verhalten wird als schwach verfestigend und zeitunabhängig angenommen. Die Normalitätsregel wird für die plastischen Verzerrungsgeschwindigkeiten eines Volumenelements postuliert. Die Ausbreitung der Wellen wird durch ein hyperbolisches System von Differentialgleichungen beschrieben, dessen Zustandsvariablen die Schnittgrößen und die Formänderungs-geschwindigkeiten eines Balkenelements sind. Schließlich wird gezeigt, daß die Einführung eines plastischen Schubkorrekturfaktors nicht erforderlich ist.

     

ROBUST CONTROL VIA STATE FEEDBACK FOR A CLASS OF UNCERTAIN BILINEAR SYSTEMS

ＲＯＢＵＳＴＣＯＮＴＲＯＬＶＩＡＳＴＡＴＥＦＥＥＤＢＡＣＫＦＯＲＡＣＬＡＳＳＯＦＵＮＣＥＲＴＡＩＮＢＩＬＩＮＥＡＲＳＹＳＴＥＭＳ￥ＣｈｅｎＳｏｎｇｌｉｎ（陈松林）；ＺｈｕＺｕｃｈｉ（朱祖慈）（ＥａｓｔＣｈｉｎａＩｎｓｔｉｔｕｔｅｏｆＭｅｔａｌｌｕｒｇ...     

The suppression of a dynamic instability of an elastic body using feedback control

A theoretical and experimental study is made to determine the feasibility of controlling a thin cantilevered beam subject to a (nonconservative) follower force. A theoretical model is developed using the equations for a thin beam under initial stress and Galerkin's method. An experiment is constructed with the capability of using a variety of feedback loops to control a thin aluminum beam with a tip jet mounted parallel to the chord. A particular control system is chosen for study and an increase of follower force required to destabilize the beam of over 65 per cent is recorded. The theoretical results show good correlation with the experimentally determined stability boundaries and frequency variations with follower force.     

Projective synchronization of a class of chaotic systems by dynamic feedback control method

This paper investigates the projective synchronization problem of a class of chaotic systems in arbitrary dimensions. Firstly, a necessary and sufficient condition for the existence of the projective synchronization problem is presented. And this condition is equivalent to check whether a group of algebraic equations about \(\alpha \) have solutions or not. Secondly, an algorithm is proposed to obtain all the solutions of the projective synchronization problem. Thirdly, a simple and physically implementable controller is designed to ensure the realization of the projective synchronization. Finally, three numerical examples are provided to verify the effectiveness and the validity of the proposed results.     

Least squares algorithm for an input nonlinear system with a dynamic subspace state space model

For a Hammerstein input nonlinear system with a subspace state space linear element, this paper transforms the system into a bilinear identification model by using the property of the shift operator to the state space model and presents a recursive and an iterative least squares algorithms to generate parameter estimates and state estimates by using the hierarchical identification principle and by replacing the unknown state variables with their estimates. The proposed approaches are computationally more efficient than the over-parameterization model based least squares method.     

An analytical study of time-delayed control of friction-induced vibrations in a system with a dynamic friction model

We investigate the control of friction-induced vibrations in a system with a dynamic friction model which accounts for hysteresis in the friction characteristics. Linear time-delayed position feedback applied in a direction normal to the contacting surfaces has been employed for the purpose. Analysis shows that the uncontrolled system loses stability via. a subcritical Hopf bifurcation making it prone to large amplitude vibrations near the stability boundary. Our results show that the controller achieves the dual objective of quenching the vibrations as well as changing the nature of the bifurcation from subcritical to supercritical. Consequently, the controlled system is globally stable in the linearly stable region and yields small amplitude vibrations if the stability boundary is crossed due to changes in operating conditions or system parameters. Criticality curve separating regions on the stability surface corresponding to subcritical and supercritical bifurcations is obtained analytically using the method of multiple scales (MMS). We have also identified a set of control parameters for which the system is stable for lower and higher relative velocities but vibrates for the intermediate ones. However, the bifurcation is always supercritical for these parameters resulting in low amplitude vibrations only.     

Stability of a two-dimensional airfoil with time-delayed feedback control

This paper presents a systematic study on aeroelastic stability of a two-dimensional airfoil with a single or multiple time delays in the feedback control loops. Firstly, the delay-independent stability region of the aeroelastic system with a single time delay is determined on the basis of the generalized Sturm criterion for polynomials. Then, the stability switches with variations in time delay are analyzed when the system parameters fall out of the delay-independent stability region. Flutter boundaries of the controlled aeroelastic system as time delay varies are predicted in a continuous way by the predictor-corrector technique. Finally, two methods, the polynomial eigenvalue method and the infinitesimal generator method, are introduced to investigate the stability of the controlled aeroelastic system with multiple time delays. Numerical simulations are made to demonstrate the effectiveness of all the above approaches.     

Adaptive output feedback control of uncertain nonlinear chaotic systems based on dynamic surface control technique

In this paper, an adaptive output feedback control algorithm based on the dynamic surface control (DSC) is proposed for a class of uncertain chaotic systems. Because the system states are assumed to be unavailable, an observer is designed to estimate those unavailable states. The main advantage of this algorithm can overcome the problem of “explosion of complexity” inherent in the backstepping design. Thus, the proposed control approach is simpler than the traditional backstepping control for the uncertain chaotic systems. The stability analysis shows that the system is stable in the sense that all signals in the closed-loop system are uniformly ultimately bounded (UUB) and the system output can track the reference signal to a bounded compact set. Finally, an example is provided to illustrate the effectiveness of the proposed control system.     

Output feedback stabilization based on dynamic surface control for a class of uncertain stochastic nonlinear systems

In this paper, the dynamic surface control (DSC) algorithm is proposed for a class of stochastic nonlinear systems with nonminimum phase and the standard output-feedback form. The proposed algorithm is a stochastic vision by combining the traditional back-stepping together with the DSC technique, which can overcome the problem of ‘explosion of complexity’ in the back-stepping designing procedure for the stochastic nonlinear systems. Thus, it can reduce the computation complexity and is easy to be used in the actual implementation. It is shown that all the signals of the resulting closed-loop system are uniformly ultimately bounded.     

Parameter design for a vibration absorber with time-delayed feedback control

Traditional passive vibration absorbers are effective only when their natural frequencies are close to those of the excitations. To solve this problem, a vibration absorber with time-delayed feedback control is proposed to suppress vibration of the primary system under excitation with changing frequency. Firstly, the mechanical model of the delay coupled system is established. Then, the displacement transfer ratio of the system is obtained. The stability of the system is analyzed since delay may result in destabilization. Next, in order to design the control parameters, the vibration absorption performances of the proposed time-delayed vibration absorber are studied. The vibration absorption region is shown. The results show that time-delayed feedback control is able to change the response of the system. The effective vibration absorption frequency band is adjustable by tuning the control gain and time delay. The effective frequency band can be widened when choosing appropriate control parameters. The vibration absorption performances can be greatly improved by the time-delayed absorber. In addition, the optimum control parameters are obtained. Finally, the experimental prototype is constructed. Several tests with different control parameters are taken. The experimental and analytical results match quite well.